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Volumn 81, Issue 6, 2010, Pages

Fundamental theory of statistical particle dynamics

Author keywords

[No Author keywords available]

Indexed keywords

BROWNIAN MOTION; CLASSICAL PARTICLE; ERGODICS; FIELD THEORY; FUNCTIONAL THEORY; FUNDAMENTAL THEORY; LIQUID-GLASS TRANSITION; NON-ERGODIC TRANSITION; PARTICLE DYNAMICS;

EID: 77953440627     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.81.061102     Document Type: Article
Times cited : (44)

References (168)
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    • In the weak supercooling regime, detailed predictions for the space and time dependence of the long-time decay of density correlations have been formulated using the ideal mode-coupling theory MCT, one of the first approaches to identify the existence of the crossover temperature. The agreement of MCT predictions with experimental findings and molecular dynamics simulations both for atomic and molecular models supports the view that MCT is indeed able to describe the slow dynamics in weak supercooled states.
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    • The important idea of an ergodic-nonergodic (ENE) transition exists independent of whether it conforms to the details of mode-coupling theory. The defining property of an ENE transition is that as a function of a control parameter there is crossover from an ergodic phase [ limt→ Gρρ (q,t ) =0 ] to a region with nonergodic kinetics [ limt→ Gρρ (q,t ) = A2 (q) >0 ]. Kinetic theory led to the early development of MCT. See for example
    • The important idea of an ergodic-nonergodic (ENE) transition exists independent of whether it conforms to the details of mode-coupling theory. The defining property of an ENE transition is that as a function of a control parameter there is crossover from an ergodic phase [lim t → G ρ ρ (q, t) = 0] to a region with nonergodic kinetics [lim t → G ρ ρ (q, t) = A 2 (q) > 0].
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    • More recent efforts to make microscopic contact (derive) mode-coupling theory are reflected in Ref. where they comment: "Despite its remarkable practical success, the presence of apparently uncontrolled approximations in the derivation of the MCT equations makes it difficult to gain insights into possible improvements of the theory. The aim of this paper is to present a new derivation of the ideal MCT equations, starting from the microscopic equations for the evolution of the density (Newtons equations) and writing them as a linear generalized Langevin equation. A formally exact expression for the memory kernel is derived and, on making the approximation that the noise in the Langevin equation is Gaussian, the standard MCT equations are obtained. Note that the proposition of Gaussian noise implies that the density fluctuations are also Gaussian."
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    • Our interest here is the theoretical work of Biroli and Bouchaud in Ref.. Biroli and Bouchaud sketched a field theoretical calculation of C 4 (r, t) compatible with mode-coupling theory. This calculation leads to a diverging length scale at the ideal glass transition. They find an upper critical dimension of six in their calculation. This work is very provocative since at the level of the dynamic structure factor it is well known that MCT does not contain a large length as one approaches the ergodic-nonergodic transition. Biroli and Bouchaud suggest that one must dig deeper into the theory, look at the four-point quantity C 4, to find this diverging length. However this calculation paints a picture rather than gives the results of a rigorous calculation. One of the goals for the theory developed here is to connect the collective behaviors found at the two-point level to the behavior of the three- and four-point density correlation functions.
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    • Purely dissipative Langevin systems have long been used to describe the order parameter dynamics in magnetic and superfluid systems. There is a balance between the force due to the gradient of the effective free energy and the persistent noise in a thermalized system.The path-integral description of classical Newtonian dynamics is developed in a series of sophisticated papers by Gozzi and collaborators.
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    • Their interest is in the symmetry structure of the coupled spaces when one allows the classical system to be coupled to a system of ghost fermions. See below.
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    • Fokker-Planck dynamics in modern terms has two complementary meanings. For systems of particles governed by phase-space coordinates and with a damping component in their dynamics the equation satisfied by the associated probability distribution is called a Fokker-Planck equation. As pointed out by Lax and Zwanzig, the associated equation of motion for the phase-space variables is referred to as a Langevin equation. Subsequently the transition generally from a stochastic equation of motion to the time evolution of the associated probability description is called going from a generalized Langevin equation description to generalized Fokker-Planck description. See Ref.. Smoluchowski dynamics has become identified with the over damped kinetics in colloidal systems where the momenta become equilibrated much faster than the positions and one has a dynamics which is subsequently organized in terms of the positions. This is the point of view taken by
    • Fokker-Planck dynamics in modern terms has two complementary meanings. For systems of particles governed by phase-space coordinates and with a damping component in their dynamics the equation satisfied by the associated probability distribution is called a Fokker-Planck equation. As pointed out by Lax and Zwanzig, the associated equation of motion for the phase-space variables is referred to as a Langevin equation. Subsequently the transition generally from a stochastic equation of motion to the time evolution of the associated probability description is called going from a generalized Langevin equation description to generalized Fokker-Planck description. See Ref..
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    • ). If the density is a Gaussian variable, then W [u] = d d x 1 d d x 2 u (x 1) G (x 1 - x 2) u (x 2).
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    • This is just the functional generalization of the standard matrix identity exp [1 2 i j K i j δ 2 δ h i δ h j] e i h i i = e i h i i exp [1 2 i j K i j i j].
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    • Quantum Thermal Green's functions must satisfy Kubo-Martin-Schwinger Boundary Conditions, see references in Ref..
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